The best racing games like Excitebike are Trials Rising, Excitebike 64 and Joe Danger. Below are 8, each with the reason it is here — and, at the end, what filtering to racing costs you.
Filtering to racing means you want the vehicle to be the subject. It is the right filter when what you liked was speed, or handling, or a road worth driving.
What you get: A short session that is complete in itself. A race is over in five minutes and you can always take the corner better.
What made Excitebike worth playing in the first place is covered on the full list. This page is the racing part of that answer, and only the racing part.
The direct descendant — physics, angles, and short punishing runs.
The 3D sequel, and better than it is remembered.
Side-on stunt racing built to be cheerful about failure.
Downhill physics racing that is all about flow.
If it was the track editor and the four-player chaos you wanted.
Bikes with the arcade attitude turned up.
Pure physics-bike puzzling, and famously hard.
Quiet, precise downhill riding with no opponents.
Everything outside the vehicle. Racing games rarely have a story, and the ones that try usually should not have.
Specifically, these are strong answers for Excitebike that are not racing games, so they are not on this page:
The full list is on games like Excitebike — 9 picks, no genre filter.
Trials Rising. The direct descendant — physics, angles, and short punishing runs.
No. Excitebike released on the Nintendo Entertainment System, but a recommendation is only useful if you can play it, so the picks above span platforms and eras. What they share is the genre and the reason.
Simulation models tyre temperature, fuel and weight transfer and punishes you for ignoring them. Arcade gives you grip, boost and a rewind, and is about flow. Buying the wrong one is where most disappointment comes from.
For the simulation picks it changes the game completely. For anything arcade it makes almost no difference.
By hand. Each pick was written for the full Excitebike list first, on what the game actually does, and this page is the part of that list that is racing — never a genre tag lookup, which is how "best racing games like Super Mario 64" used to be a page on this site.
Or read the racing guide for where to start in the genre generally.